Path Eccentricity of k-AT-Free Graphs and Consecutive Ones Property Relationship

Основные понятия

The author explores the relationship between path eccentricity in k-AT-free graphs and the consecutive ones property, providing insights into their bounds and connections.

Аннотация

The content delves into the central path problem in graph theory, focusing on path eccentricity in k-AT-free graphs. It establishes a link between path eccentricity and the consecutive ones property, offering new insights and generalizations. Theorems are proven to show bounds on path eccentricity based on graph properties.
Key points include defining k-AT-free graphs, proving theorems related to path eccentricity bounds, discussing the consecutive ones property, and exploring implications for graph structures. Lemmas are used to demonstrate ordering constraints in induced paths and cycles within graphs with the *-C1P.
The content concludes by showing that graphs with the *-C1P do not contain 2-asteroidal triples, leading to a maximum path eccentricity of 2. This result is illustrated through detailed proofs and references to related works.

Статистика

Theorem 1 ([COS97]): If G is an AT-free graph, then pe(G) ≤ 1.
Theorem 2 ([GG23]): Let G = (X ∪ Y, E) be a bipartite graph. If G is X-convex then pe(G) ≤ 2. If G is also Y -convex (i.e., biconvex), then pe(G) ≤ 1.
Theorem 3: For every k ≥ 1, if a graph G is k-AT-free then pe(G) ≤ k.

Цитаты

"Graphs whose adjacency matrices have the consecutive ones property have path eccentricity at most 1." - Content
"If a graph has the *-C1P then it has no 2-AT." - Content
"The results widen the knowledge of the path eccentricity of graphs as summarized." - Content

Ключевые выводы из

Path eccentricity of $k$-AT-free graphs and application on graphs with the consecutive ones property

by Paul Bastide... в arxiv.org 03-11-2024

https://arxiv.org/pdf/2403.05360.pdf

Дополнительные вопросы

How does the concept of path eccentricity in graphs extend beyond theoretical applications

Path eccentricity in graphs extends beyond theoretical applications by providing insights into the structural properties of networks. Understanding path eccentricity can help in identifying central paths or routes that minimize the distance between various nodes in a network. This concept is crucial in optimizing transportation systems, logistics planning, and communication networks. By determining the path eccentricity of a graph, one can efficiently locate key facilities or hubs to improve connectivity and reduce overall travel time within the network.

What potential real-world scenarios could benefit from understanding these properties in practical settings

Real-world scenarios that could benefit from understanding path eccentricity and related properties include urban planning for designing efficient road networks, emergency response management for establishing optimal routes for first responders, and supply chain management to streamline distribution channels. Additionally, telecommunications companies can use this knowledge to enhance signal propagation paths and coverage areas. By applying these concepts practically, organizations can improve resource allocation, reduce operational costs, and enhance overall system performance.

How might advancements in this area impact algorithm development for network optimization

Advancements in understanding path eccentricity in graphs have significant implications for algorithm development in network optimization. By leveraging insights from path eccentricity analysis, researchers can design more efficient algorithms for tasks such as facility location optimization, route planning algorithms (e.g., Dijkstra's algorithm), network flow optimization (e.g., maximum flow algorithms), and clustering techniques based on centrality measures. These advancements enable faster computation of optimal paths or configurations within complex networks, leading to improved decision-making processes across various industries like transportation, telecommunications, and infrastructure development.